1. Introduction Graphene Bulk-edge correspondence Results
The Colors of Graphene:
Hofstadter Butterfly for the Honeycomb Lattice
Andrea Agazzi, Gian Michele Graf, Jean-Pierre Eckmann
Département de Physique Théorique
Zurich, 14.10.2014
3. Introduction Graphene Bulk-edge correspondence Results
The Hofstadter model (the square lattice)
The Hamiltonian
H =
4
j=1
Tj .
acting on
H = 2
(Z2
; C)
(m, n)
(m, n + 1)
(m + 1, n)
Explicitly
Hψm,n = ψm−1,n + ψm+1,n
e−2πimΦ/Φ0
ψm,n−1 + e2πimΦ/Φ0
ψm,n+1
Assumption: Rational magnetic field flux per unit cell
Φ/Φ0 = p/q with p, q ∈ N .
5. Introduction Graphene Bulk-edge correspondence Results
The Diophantine equation
Proposition [Thouless et al., 1982, Dana et al., 1985]:
The Hall conductivity σH in the r-th spectral gap of H is the
solution of the Diophantine equation
r = σH · p + s · q
where s ∈ Z.
Remark (the window condition):
It is natural to impose the uniqueness condition
σH ∈ (−q/2, q/2) .
7. Introduction Graphene Bulk-edge correspondence Results
Lattice structure
BA
m, n
m, n + 1
m + 1, n
Coordinate system on
the honeycomb lattice
structure.
Hexagonal lattice structure
Bipartite lattice
Spinor-like wave function
ψm,n =
ψA
m,n
ψB
m,n
∈ C2
ψ = (ψm,n)m,n ∈ H = 2
(Z2
; C2
)
9. Introduction Graphene Bulk-edge correspondence Results
The Diophantine equation
The proof of [Thouless et al., 1982, Dana et al., 1985] shows that
also for the hexagonal lattice the value of σH in the r-th gap must
satisfy
r = σH · p + s · q .
How about the window condition?
Can the natural window condition still be applied?
12. Introduction Graphene Bulk-edge correspondence Results
The edge lattice
New physical space: the half plane lattice.
H = 2
(N × Z; C2
)
The new Hamiltonian H is the restriction
of H to the Hilbert space H.
Bloch decomposition in the unbroken
symmetry direction gives H(k):
ψB
1 (k)
ψA
1 (k)
(Hψ)m(k) =
ψB
m(k) + ψB
m+1(k) + e−2πimp/q+ik
ψB
m(k)
ψA
m(k) + ψA
m−1(k) + e2πimp/q−ik
ψA
m(k)
13. Introduction Graphene Bulk-edge correspondence Results
The bulk-edge correspondence [Hatsugai, 1993]
Edge spectrum E(k) (red lines):
E
EF
k
2π0
Theorem:
# of (signed) crossings of the E(k) in k ∈ (0, 2π) = σH(r).
14. Introduction Graphene Bulk-edge correspondence Results
The transfer operator formalism
Define the transfer operators on the half-plane at the energy E and
wave-vector k as
ψB
m+1(k)
ψA
m(k)
= T E
m (k)
ψB
m(k)
ψA
m−1(k)
.
and a translation of q dimers by
T E
(k) =
q
m=1
T E
m (k) = T E
q (k) · · · T E
1 (k) .
The latter operator allows us to describe the wave function on the
whole lattice given its value at any two neighboring points in the
lattice.
15. Introduction Graphene Bulk-edge correspondence Results
The detection of edge states
An edge state wave function must satisfy:
1 Boundedness
Be normalizable, i.e. be a contracting
eigenvector of T E
(k).
2 Edge
Vanish to the left of the boundary i.e.,
ψB
1 (k)
ψA
0 (k)
∼
1
0
ψB
1 (k)
ψA
1 (k)
Lemma [A., Eckmann and Graf, 2014]
Let (a(k), b(k)) be the contracting eigenvector of T E
(k), then
σH =
2π
0
dk
2πi
∂
∂k
log
a(k) + ib(k)
a(k) − ib(k)
=
θ(2π) − θ(0)
2π
16. Introduction Graphene Bulk-edge correspondence Results
The detection of crossings
0
θ(t)
2π
-1
-2
-3
-4
-5
-6
π 2π
k
Evolution of the phase θ(k)/2π (on the y-axis) for two different
discretizations as a function of k ∈ [0, 2π) (on the x-axis) for values
of p/q = 8/19, r = 1.
The red curve has a lower discretization than the blue one.
17. Introduction Graphene Bulk-edge correspondence Results
The detection of crossings
The result must still satisfy the Diophantine equation
r = σH · p + s · q . (1)
Test:
For each value of p, q, r insert the computed value of σH into (1). If
s is “close to” a natural number, our guess can be corrected.
18. Introduction Graphene Bulk-edge correspondence Results
Results
Statistics of results:
right 99.8%
correctable 0.1%
not correctable 0.1%
of image pixels.
19. Introduction Graphene Bulk-edge correspondence Results
The Conjecture
Conjecture: [A., Eckmann and Graf, 2014]
For the hexagonal lattice, σH in the r-th gap must satisfy
r = σH · p + s · q ,
where s ∈ Z under the relaxed condition
σH ∈ (−q, q) .
Question: can this ambiguity be solved algebraically?
21. Introduction Graphene Bulk-edge correspondence Results
References
Agazzi, A., Eckmann, J.-P., and Graf, G. (2014).
The colored hofstadter butterfly for the honeycomb lattice.
Journal of Statistical Physics, pages 1–10.
Avila, J. C., Schulz-Baldes, H., and Villegas-Blas, C. (2013).
Topological invariants of edge states for periodic two-dimensional models.
Mathematical Physics, Analysis and Geometry, 16:137–170.
Dana, I., Avron, Y., and Zak, J. (1985).
Quantised Hall conductance in a perfect crystal.
J. Phys. C, 18(22):L679.
Hatsugai, Y. (1993).
Edge states in the integer quantum hall effect and the riemann surface of the bloch function.
Phys. Rev. B, 48:11851–11862.
Thouless, D. J., Kohmoto, M., Nightingale, M. P., and den Nijs, M. (1982).
Quantized Hall conductance in a two-dimensional periodic potential.
Phys. Rev. Lett., 49:405–408.
